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(b)    Deduce from (a) that there are the following three alternatives : either the series
(ane~^ns) does not converge for any value of s, or it converges for all values of s, or
else there exists a real number cr0 such that the series converges for &s > a0 and does
not converge when &s < cr0 - In the first (resp. second) case we put cr0 = +00 (resp.
— oo). In all cases, the (extended real) number cr0 is called the abscissa of convergence
of the series. The sum of the series is an analytic function of s in the region %s > a0 .

(c)    Show that if o-0 ^0, then <TO = lim sup (log |S0>n|)/An. (Show first that if the


series with general term ane~*nS — bn is convergent (with s real and >0), then we
have |S0,ii| ^ K>A"S where K is a constant, by writing an = bne*"s. Then show that if
y = lim sup (log |S0, n|)/An, the series (ane~*nS) converges when s = y + S, where 8 is


real and >0, by arguing as in (a).)

(d)    Let CTI be the abscissa of convergence of the Dirichlet series with general term
\an\e~*nS. Then a± ^ o-0 . If a0 < + oo, show that

^ ,.           1°SW

(TI — o"0 ^ lim sup — r — .

n-*oo        An

(Remark that, given e > 0, we have \an\ <; e(<T°+e)An for all sufficiently large n.) Consider
the case where An = n (cf. Section 9.1 , Problem 1). Show that, for the series with
general term


: _ IL. g-siogiogn

we have a0 — — oo and crt = +00.

10.   Let/be a real-valued function defined on the interval [0, + oo [, satisfying the following

(D   /(J+0£/(*)+/(0;

(2)   there exists a number M > 0 such that |/(r)| ^ M/ for all /.
Show that the limits

r                R    r     /(O

lim — ,      p = lim  -1-

t-*0     t                     t-t' + oo     t

exist and are finite, and that a/ <i/(0 < fit for all t ;> 0. (To establish the existence
of a it is enough to show that, if we take a — lim sup /(/)// (Problem 8), then


a ^f(t)/t for all t > 0. Take a decreasing sequence (tn) tending to 0, such that
\imf(tn)/tn = a, and let kn be the integral part of tjtn . Show that

The proof of the existence of lim f(t)/t is similar; put ft = lim ii

t-* + °o                                                            r-* + oo

11.   Let/be a continuous mapping of an open set U in a Banach space E into a Banach
space F. For each x e U, let


y-*,y**         y~

We have 0 g D~/(x)For each